HPAS 2016 Maths Optional Paper-1 Question 1(a)
Determine all values of d for which rank of the matrix: \[ \begin{pmatrix} d & -1 & 0 & 0 \\ 0 & d & -1 & 0 \\ 0 & 0 & d & -1 \\ -6 & 11 & -6 & 1 \end{pmatrix} \] is equal to 3.For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 1(b)
Show that the vectors \(v_1=(1,1,2,4)\), \(v_2 = (2, -1, -5, 2)\), \(v_3=(1,-1,-4,0)\) and \(v_4=(2,1,1,6)\) are linearly dependent in \(\mathbb{R}^4\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 1(c)
Evaluate: \[ \lim_{x\to 0} \left(\frac{\tan x}{x}\right)^{1/x^2} \]For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 1(d)
Find the angle between the surfaces: \(x \log z = y^2 – 1\) and \(x^2y = 2 – z\) at point (1, 1, 1).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 1(e)
Show that the Legendre polynomial \(P_n(x)\) satisfies \(P_n(-x)=(-1)^n P_n(x)\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 2(a)
Let T be a linear operator on \(\mathbb{R}^3\) defined by \(T(x,y,z)=(3x, x-y, 2x+y+z)\). Show that T is invertible and determine \(T^{-1}\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 2(b)
Determine the value of a and b so that the system of equations: \[ \begin{cases} 2x+3y+5z=9 \\ 7x+3y-2z=8 \\ 2x+3y+az=b \end{cases} \] has: (i) no solution (ii) a unique solution (iii) an infinite number of solutions.For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 3(a)
Show that the function f defined on \(\mathbb{R}\) by: \[ f(x) = \begin{cases} x, & \text{if x is irrational} \\ -x, & \text{if x is rational} \end{cases} \] is continuous only at \(x=0\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 3(b)
Find the asymptotes of the curve: \(2x^3 – 5x^2y + 4xy^2 – y^3 + 6x^2 – 7xy + y^2 – x + 5y – 3 = 0\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 4(a)
Find the extreme values of \(f(x,y,z)=2x+3y+z\) such that \(x^2+y^2=5\) and \(x+z=1\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 4(b)
Evaluate the integral: \[ \int_{0}^{2}\int_{0}^{y^2/2}\frac{y}{\sqrt{x^2+y^2+1}}dxdy \]For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 5(a)
Determine the general and singular solution of the non-linear differential equation: \(y = xy’ + (y’)^2\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 5(b)
Solve the differential equation: \[ (D^2 – 2D + 2)y = e^x \tan x \] where \(D = \frac{d}{dx}\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 6(a)
Find the directional derivative of the scalar function \(\phi = xy^2 + yz^3\) at point (2, -1, 1) in the direction of the normal to the surface \(x \log z – y^2 = -4\) at point (-1, 2, 1).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 6(b)
Show that the field of force given by: \(\vec{F} = (y^2 \cos x + z^3)\mathbf{i} + (2y \sin x – 4)\mathbf{j} + (3xz^2 + 2)\mathbf{k}\) is conservative and find the work done in moving the particle in the field from a point A (0, 1, 1) to a point B (\(\pi/2\), -1, 2).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 7(a)
Show that the equation: \(2x^2 – 6y^2 – 12z^2 + 18yz + 2zx + xy = 0\) represents a pair of planes and find the angle between them.For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 7(b)
Find the equations of the tangent planes to the hyperboloid \(2x^2 – 6y^2 + 3z^2 = 5\) which pass through the lines \(3x-3y+6z-5=0\) and \(x+9y-3z=0\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 8(a)
Let V be a finite dimensional inner product space over field F, and let \(g: V \to F\) be a linear transformation. Then show that there exists a unique vector \(y \in V\) such that \(g(x) = \langle x,y \rangle\) for all \(x \in V\).For solution: Click here
HPAS 2016 Maths Optional Paper-1 Question 8(b)
A particle moves in a plane in such a manner that its tangential and normal accelerations are always equal and its velocity varies as \(e^{\tan^{-1}(s/c)}\), s being the length of the arc of the curve measured from a fixed point on the curve. Find the path.For solution: Click here